# sketch in a Complex plane

$f:\mathbb{C} \to \mathbb{C}$ and $f(z)=2iz^3+i$
$A$={${z \in \mathbb{C}:Re(z)>0, Im(z)<0}$}

find and sketch $f(A)$

I tried to do it like that $$f(z)=2i|z|^3(\cos\alpha+i\sin\alpha)^3+i$$ $$=2i|z|^3(\cos(3\alpha)+i\sin(3\alpha)+i$$$$=2i|z|^3\cos(3\alpha)-|z|^32\sin(3\alpha)+i$$ $$\left\{ \begin{array} -2|z|^3\sin(3\alpha)>0 \\2|z|^3\cos(3\alpha)+1<0 \end{array} \right.$$

What should i do now sketch both of this and find an intersection? how to do it?

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You are asked to find how the image of $f$ under $A$ looks like. What kind of shape does $A$ have in the complex plane? What's $f(0)$? How does $f(z)$ change if you start at $0$ and vary $z$ along one of the borders of $A$? (eg consider purely imaginary or purely real values of $z$) –  Roland Feb 14 '14 at 20:07